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January  2020, 16(1): 309-324. doi: 10.3934/jimo.2018153

On the M-eigenvalue estimation of fourth-order partially symmetric tensors

Haitao Che 1,, , Haibin Chen 2, and  Yiju Wang 2,

1. 

School of Mathematics and Information Science, Weifang University, Weifang Shandong, 261061, China
2. 

School of Management Science, Qufu Normal University, Rizhao Shandong, 276826, China

* Corresponding author: Haitao Che

Received  January 2018 Revised  May 2018 Published  January 2020 Early access  September 2018

Fund Project: This project is supported by the Natural Science Foundation of China (11401438, 11671228, 11601261, 11571120), Shandong Provincial Natural Science Foundation (ZR2016AQ12), Project of Shandong Province Higher Educational Science and Technology Program(Grant No. J14LI52), and China Postdoctoral Science Foundation (Grant No. 2017M622163, 2018T110669).

In this article, the M-eigenvalue of fourth-order partially symmetric tensors is estimated by choosing different components of M-eigenvector. As an application, some upper bounds for the M-spectral radius of nonnegative fourth-order partially symmetric tensors are discussed, which are sharper than existing upper bounds. Finally, numerical examples are reported to verify the obtained results.

Keywords: M-eigenvalue, Z-eigenvalue, fourth-order partially symmetric tensors, spectral property, localization theorem, nonnegative tensors.
Mathematics Subject Classification: Primary: 15A06, 74B20; Secondary: 47J25.
Citation: Haitao Che, Haibin Chen, Yiju Wang. On the M-eigenvalue estimation of fourth-order partially symmetric tensors. Journal of Industrial and Management Optimization, 2020, 16 (1) : 309-324. doi: 10.3934/jimo.2018153
References:
[1]

K. Chang, K. Pearson and T. Zhang, Some variational principles for Z-eigenvalues of nonnegative tensors, Linear Algebra Appl., 438 (2013), 4166-4182.  doi: 10.1016/j.laa.2013.02.013.

[2]

H. Chen, Z. Huang and L. Qi, Copositivity detection of tensors: Theory and algorithm, J. Optimiz. Theory Appl., 174 (2017), 746-761.  doi: 10.1007/s10957-017-1131-2.

[3]

H. Chen, Y. Chen, G. Li and L. Qi, A semi-definite program approach for computing the maximum eigenvalue of a class of structured tensors and its applications in hypergraphs and copositivity test, Numer. Linear Algebra Appl., 25 (2018), e2125, 16pp. doi: 10.1002/nla.2125.

[4]

H. Chen, Z. Huang and L. Qi, Copositive tensor detection and its applications in physics and hypergraphs, Comput. Optim. Appl., 69 (2018), 133-158.  doi: 10.1007/s10589-017-9938-1.

[5]

H. Chen and Y. Wang, On computing minimal H-eigenvalue of sign-structured tensors, Front. Math. China, 12 (2017), 1289-1302.  doi: 10.1007/s11464-017-0645-0.

[6]

H. Chen, L. Qi and Y. Song, Column sufficient tensors and tensor complementarity problems, Front. Math. China, 13 (2018), 255-276.  doi: 10.1007/s11464-018-0681-4.

[7]

S. Chirit$\check{a}$, A. Danescu and M. Ciarletta, On the srtong ellipticity of the anisotropic linearly elastic materials, J. Elasticity, 87 (2007), 1-27.  doi: 10.1007/s10659-006-9096-7.

[8]

B. Dacorogna, Necessary and sufficient conditions for strong ellipticity for isotropic functions in any dimension, Discrete Cont. Dyn-B, 1 (2001), 257-263.  doi: 10.3934/dcdsb.2001.1.257.

[9]

W. Ding, J. Liu, L. Qi and H. Yan, Elasticity M-tensors and the strong ellipticity condition, preprint, arXiv: 1705.09911.

[10]

D. Han, H. Dai and L. Qi, Conditions for strong ellipticity of anisotropic elastic materials, J. Elasticity, 97 (2009), 1-13.  doi: 10.1007/s10659-009-9205-5.

[11]

J. He and T. Huang, Upper bound for the largest Z-eigenvalue of positive tensors, Appl. Math. Lett., 38 (2014), 110-114.  doi: 10.1016/j.aml.2014.07.012.

[12]

Z. Huang and L. Qi, Positive definiteness of paired symmetric tensors and elasticity tensors, J. Comput. Appl. Math., 338 (2018), 22-43.  doi: 10.1016/j.cam.2018.01.025.

[13]

J. K. Knowles and E. Sternberg, On the ellipticity of the equations of non-linear elastostatics for a special material, J. Elasticity, 5 (1975), 341-361.  doi: 10.1007/BF00126996.

[14]

J. K. Knowles and E. Sternberg, On the failure of ellipticity of the equations for finite elastostatic plane strain, Arch. Rational Mech. Anal., 63 (1977), 321-336.  doi: 10.1007/BF00279991.

[15]

E. Kofidis and P. Regalia, On the best rank-1 approximation of higher-order supersymmetric tensors, SIAM J. Matrix Anal. Appl., 23 (2002), 863-884.  doi: 10.1137/S0895479801387413.

[16]

C. Padovani, Strong ellipticity of transversely isotropic elasticity tensors, Meccanica, 37 (2002), 515-525.  doi: 10.1023/A:1020946506754.

[17]

L. Qi, H. Dai and D. Han, Conditions for strong ellipticity and M-eigenvalues, Front. Math. China, 4 (2009), 349-364.  doi: 10.1007/s11464-009-0016-6.

[18]

L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symb. Comput., 40 (2005), 1302-1324.  doi: 10.1016/j.jsc.2005.05.007.

[19]

Y. Song and L. Qi, Spectral properties of positively homogeneous operators induced by higher order tensors, SIAM J. Matrix Anal. Appl., 34 (2013), 1581-1595.  doi: 10.1137/130909135.

[20]

J. R. Walton and J. P. Wilber, Sufficient conditions for strong ellipticity for a class of anisotropic materials, Int. J. Nonlin. Mech., 38 (2003), 441-455.  doi: 10.1016/S0020-7462(01)00066-X.

[21]

Y. Wang, L. Caccetta and G. Zhou, Convergence analysis of a block improvement method for polynomial optimization over unit spheres, Numer. Linear Algebra and Appl., 22 (2015), 1059-1076.  doi: 10.1002/nla.1996.

[22]

X. Wang, H. Chen and Y. Wang, Solution structures of tensor complementarity problem, Front. Math. China, 13 (2018), 935-945.  doi: 10.1007/s11464-018-0675-2.

[23]

Y. Wang, L. Qi and X. Zhang, A practical method for computing the largest M-eigenvalue of a fourth-order partially symmetric tensor, Numer. Linear Algebra Appl., 16 (2009), 589-601.  doi: 10.1002/nla.633.

[24]

Y. Wang, K. Zhang and H. Sun, Criteria for strong H-tensors, Front. Mathe. China, 11 (2016), 577-592.  doi: 10.1007/s11464-016-0525-z.

[25]

G. Wang, G. Zhou and L. Caccetta, Z-eigenvalue inclusion theorems for tensors, Discrete Cont. Dyn-B, 22 (2017), 187-198.  doi: 10.3934/dcdsb.2017009.

[26]

T. Zhang and G. Golub, Rank-1 approximation of higher-order tensors, SIAM J. Matrix Anal. Appl., 23 (2001), 534-550.  doi: 10.1137/S0895479899352045.

[27]

K. Zhang and Y. Wang, An H-tensor based iterative scheme for identifying the positive definiteness of multivariate homogeneous forms, J. Comput. Appl. Math., 305 (2016), 1-10.  doi: 10.1016/j.cam.2016.03.025.

[28]

G. Zhou, G. Wang, L. Qi and M. Alqahtani, A fast algorithm for the spectral radii of weakly reducible nonnegative tensors, Numer. Linear Algebra Appl., 25 (2018), e2134, 10pp. doi: 10.1002/nla.2134.

show all references

References:
[1]

K. Chang, K. Pearson and T. Zhang, Some variational principles for Z-eigenvalues of nonnegative tensors, Linear Algebra Appl., 438 (2013), 4166-4182.  doi: 10.1016/j.laa.2013.02.013.

[2]

H. Chen, Z. Huang and L. Qi, Copositivity detection of tensors: Theory and algorithm, J. Optimiz. Theory Appl., 174 (2017), 746-761.  doi: 10.1007/s10957-017-1131-2.

[3]

H. Chen, Y. Chen, G. Li and L. Qi, A semi-definite program approach for computing the maximum eigenvalue of a class of structured tensors and its applications in hypergraphs and copositivity test, Numer. Linear Algebra Appl., 25 (2018), e2125, 16pp. doi: 10.1002/nla.2125.

[4]

H. Chen, Z. Huang and L. Qi, Copositive tensor detection and its applications in physics and hypergraphs, Comput. Optim. Appl., 69 (2018), 133-158.  doi: 10.1007/s10589-017-9938-1.

[5]

H. Chen and Y. Wang, On computing minimal H-eigenvalue of sign-structured tensors, Front. Math. China, 12 (2017), 1289-1302.  doi: 10.1007/s11464-017-0645-0.

[6]

H. Chen, L. Qi and Y. Song, Column sufficient tensors and tensor complementarity problems, Front. Math. China, 13 (2018), 255-276.  doi: 10.1007/s11464-018-0681-4.

[7]

S. Chirit$\check{a}$, A. Danescu and M. Ciarletta, On the srtong ellipticity of the anisotropic linearly elastic materials, J. Elasticity, 87 (2007), 1-27.  doi: 10.1007/s10659-006-9096-7.

[8]

B. Dacorogna, Necessary and sufficient conditions for strong ellipticity for isotropic functions in any dimension, Discrete Cont. Dyn-B, 1 (2001), 257-263.  doi: 10.3934/dcdsb.2001.1.257.

[9]

W. Ding, J. Liu, L. Qi and H. Yan, Elasticity M-tensors and the strong ellipticity condition, preprint, arXiv: 1705.09911.

[10]

D. Han, H. Dai and L. Qi, Conditions for strong ellipticity of anisotropic elastic materials, J. Elasticity, 97 (2009), 1-13.  doi: 10.1007/s10659-009-9205-5.

[11]

J. He and T. Huang, Upper bound for the largest Z-eigenvalue of positive tensors, Appl. Math. Lett., 38 (2014), 110-114.  doi: 10.1016/j.aml.2014.07.012.

[12]

Z. Huang and L. Qi, Positive definiteness of paired symmetric tensors and elasticity tensors, J. Comput. Appl. Math., 338 (2018), 22-43.  doi: 10.1016/j.cam.2018.01.025.

[13]

J. K. Knowles and E. Sternberg, On the ellipticity of the equations of non-linear elastostatics for a special material, J. Elasticity, 5 (1975), 341-361.  doi: 10.1007/BF00126996.

[14]

J. K. Knowles and E. Sternberg, On the failure of ellipticity of the equations for finite elastostatic plane strain, Arch. Rational Mech. Anal., 63 (1977), 321-336.  doi: 10.1007/BF00279991.

[15]

E. Kofidis and P. Regalia, On the best rank-1 approximation of higher-order supersymmetric tensors, SIAM J. Matrix Anal. Appl., 23 (2002), 863-884.  doi: 10.1137/S0895479801387413.

[16]

C. Padovani, Strong ellipticity of transversely isotropic elasticity tensors, Meccanica, 37 (2002), 515-525.  doi: 10.1023/A:1020946506754.

[17]

L. Qi, H. Dai and D. Han, Conditions for strong ellipticity and M-eigenvalues, Front. Math. China, 4 (2009), 349-364.  doi: 10.1007/s11464-009-0016-6.

[18]

L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symb. Comput., 40 (2005), 1302-1324.  doi: 10.1016/j.jsc.2005.05.007.

[19]

Y. Song and L. Qi, Spectral properties of positively homogeneous operators induced by higher order tensors, SIAM J. Matrix Anal. Appl., 34 (2013), 1581-1595.  doi: 10.1137/130909135.

[20]

J. R. Walton and J. P. Wilber, Sufficient conditions for strong ellipticity for a class of anisotropic materials, Int. J. Nonlin. Mech., 38 (2003), 441-455.  doi: 10.1016/S0020-7462(01)00066-X.

[21]

Y. Wang, L. Caccetta and G. Zhou, Convergence analysis of a block improvement method for polynomial optimization over unit spheres, Numer. Linear Algebra and Appl., 22 (2015), 1059-1076.  doi: 10.1002/nla.1996.

[22]

X. Wang, H. Chen and Y. Wang, Solution structures of tensor complementarity problem, Front. Math. China, 13 (2018), 935-945.  doi: 10.1007/s11464-018-0675-2.

[23]

Y. Wang, L. Qi and X. Zhang, A practical method for computing the largest M-eigenvalue of a fourth-order partially symmetric tensor, Numer. Linear Algebra Appl., 16 (2009), 589-601.  doi: 10.1002/nla.633.

[24]

Y. Wang, K. Zhang and H. Sun, Criteria for strong H-tensors, Front. Mathe. China, 11 (2016), 577-592.  doi: 10.1007/s11464-016-0525-z.

[25]

G. Wang, G. Zhou and L. Caccetta, Z-eigenvalue inclusion theorems for tensors, Discrete Cont. Dyn-B, 22 (2017), 187-198.  doi: 10.3934/dcdsb.2017009.

[26]

T. Zhang and G. Golub, Rank-1 approximation of higher-order tensors, SIAM J. Matrix Anal. Appl., 23 (2001), 534-550.  doi: 10.1137/S0895479899352045.

[27]

K. Zhang and Y. Wang, An H-tensor based iterative scheme for identifying the positive definiteness of multivariate homogeneous forms, J. Comput. Appl. Math., 305 (2016), 1-10.  doi: 10.1016/j.cam.2016.03.025.

[28]

G. Zhou, G. Wang, L. Qi and M. Alqahtani, A fast algorithm for the spectral radii of weakly reducible nonnegative tensors, Numer. Linear Algebra Appl., 25 (2018), e2134, 10pp. doi: 10.1002/nla.2134.

Figure 1.  The comparisons of $ \Gamma(\mathcal{C})$, $ \mathcal{L}(\mathcal{C})$, $ \mathcal{M}(\mathcal{C})$ and $ \mathcal{N}(\mathcal{C})$
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Figure 2.  The comparisons of $ \Gamma(\mathcal{C})$, $ \mathcal{L}(\mathcal{C})$ and $ \mathcal{M}(\mathcal{C})$
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Figure 3.  The comparisons of $ \Gamma(\mathcal{C})$, $ \mathcal{L}(\mathcal{C})$ and $ \mathcal{M}(\mathcal{C})$
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